Properties

Label 51870bc
Number of curves 8
Conductor 51870
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("51870.bd1")
sage: E.isogeny_class()

Elliptic curves in class 51870bc

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.bd7 51870bc1 [1, 0, 1, -55002084, 157000947682] 6 5308416 \(\Gamma_0(N)\)-optimal
51870.bd6 51870bc2 [1, 0, 1, -55935204, 151397748706] 12 10616832  
51870.bd5 51870bc3 [1, 0, 1, -77910099, 14043169966] 2 15925248  
51870.bd8 51870bc4 [1, 0, 1, 47772876, 648698733922] 6 21233664  
51870.bd4 51870bc5 [1, 0, 1, -174573204, -704504238494] 6 21233664  
51870.bd2 51870bc6 [1, 0, 1, -832884819, -9218693675858] 4 31850496  
51870.bd3 51870bc7 [1, 0, 1, -432480339, -18098543909714] 2 63700992  
51870.bd1 51870bc8 [1, 0, 1, -13312884819, -591230981675858] 2 63700992  

Rank

sage: E.rank()

The elliptic curves in class 51870bc have rank \(1\).

Modular form None

sage: E.q_eigenform(10)
\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + q^{13} - q^{14} - q^{15} + q^{16} + 6q^{17} - q^{18} + q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.